Singular hermitian metrics on holomorphic vector bundles

Artikel i vetenskaplig tidskrift, 2015

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Pun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature I similar to (h) as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to I similar to (h) . Finally we define what it means for h to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.